School of Aerospace Engineering Independent Variations of TD Props. • Motivation – Recall, a TD state is a unique (thermodynamic) description of a substance – identified by state variables • e.g., p, T, ρ, … intensive; m, V, …. extensive) – How many TD properties must be known/specificed to uniquely define the state of a system? • Empirical evidence (experience) suggests – number of ways one can independently vary the energy of a given substance gives the number of independent TD properties State Postulate -1 Copyright © 2009 by Jerry M. Seitzman. All rights reserved. AE/ME 6765 School of Aerospace Engineering Reversible Work Modes • What are independent ways to transfer energy? – Reversible work modes + Heat transfer • What are reversible work modes? force generalized displacement δW ≡ Fdx – work mode is reversible if • F is indep. of direction and rate of change of process or • equivalent amount of energy transferred to system when x is increased by dx will be exactly the same amount of energy transferred from system when x is decreased by dx State Postulate -2 Copyright © 2009 by Jerry M. Seitzman. All rights reserved. AE/ME 6765 1 School of Aerospace Engineering Examples of Rev. Work Modes Mode dW (+ into system) Fluid Compression Polarization Liquid Surface Extension − pdV r r E ⋅ d vP ( ) σdA • Simple substance ⇒ only one reversible work mode • Simple compressible substance ⇒ only rev. work mode is fluid compression • Also, effect of any irreversible work modes can always be accomplished by combination of reversible work + heat transfer AE/ME 6765 State Postulate -3 Copyright © 2009 by Jerry M. Seitzman. All rights reserved. School of Aerospace Engineering State Postulate • Formal statement – “The number of independent, intensive TD properties of a specified substance is equal to the number of relevant reversible work modes (n) plus one.” • specified substance ⇒ must know fraction of each constituent in a mixture of substances (phases,…) • relevant modes ⇒ only care about rev. work modes for system in question • Does not indentify which set of n+1 prop’s. are indep. • Relationships between TD properties given by equations of state – generally, In+2 ≡ In+2(I1, I2,…In+1) – e.g., p=p(ρ,T) State Postulate -4 Copyright © 2009 by Jerry M. Seitzman. All rights reserved. AE/ME 6765 2 School of Aerospace Engineering Example: Simple Compressible Mixture • Assume mixture of m pure (single-phase) substances – how many properties are independent? • To specify substance, we need m-1 mole fractions χi – χi ≡ ni/ntot where ni is the number of moles of pure substance i (ni is extensive, χi is intensive) m m i =1 i =1 – ∑ ni = ntot ; ∑ χ i = 1 • so only m-1 mole fractions are independent AE/ME 6765 State Postulate -5 Copyright © 2009 by Jerry M. Seitzman. All rights reserved. School of Aerospace Engineering Example (con’t) • From state postulate we need n+1=2 additional independent intensive variables to define a state – I3 ≡ I3(I1, I2, χ1,…χm-1) • To determine any extensive property of the system, we must add one additional extensive prop. (e.g., mass) to the list – EJ ≡ EJ(E1, I1, I2, χ1,…χm-1) State Postulate -6 Copyright © 2009 by Jerry M. Seitzman. All rights reserved. AE/ME 6765 3 School of Aerospace Engineering Extensive and Intensive Props. • We can create intensive properties from extensive properties • For example, given two systems in TD equilibrium ⇒ IJ(1)=IJ(2)=IJ(3) – by additive property of extensive var. ( ( (3 ) ) () ) ( 1 2 (2 ) 3 = f M (3 ) , I1 , I 2 = E J + E J = f M (1) , I1 , I 2 + f M (2 ) , I1 , I 2 – but if this is true, then we must be able to write f (M , I1 , I 2 ) = Mg (I1 , I 2 ) E ∴ J = g (I1 , I 2 ) a (mass) specific property M AE/ME 6765 EJ 1 ) State Postulate -7 Copyright © 2009 by Jerry M. Seitzman. All rights reserved. School of Aerospace Engineering Specific Properties EJ = g ( I1 , I 2 ) M • Are intensive because they are functions ONLY of intensive properties • Examples – h≡H/M specific enthalpy – v≡V/M specific volume • Can define other types of specific props. – e.g., H/V volume specific enthalpy in general, ratios of exten. props. are intensive State Postulate -8 Copyright © 2009 by Jerry M. Seitzman. All rights reserved. AE/ME 6765 4